HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

Use quantifiers to convert the following open sentences defined on N, into a true statement.

n² ≥ 1

Check whether the following matrix is invertible or not:

`[(cos theta, sin theta),(-sin theta, cos theta)]`

Express the following equations in matrix form and solve them by the method of reduction:

x + 2y + z = 8, 2x + 3y – z = 11, 3x – y – 2z = 5.

In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot  C/2`.

Select the correct option from the given alternatives:

In ΔABC if c² + a² – b² = ac, then ∠B = ____.

The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______

With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`

Find the principal solutions of tan x = `-sqrt(3)`

In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.

If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a², b², c² are in A.P.

Find the separate equations of the lines represented by the equation 3x² – 10xy – 8y² = 0.

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax² + 2hxy + by² = 0 

Find the value of k if the lines represented by kx² + 4xy – 4y² = 0 are perpendicular to each other. 

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0` 

If the foot of the perpendicular drawn from the origin to the plane is (4, −2, -5), then the equation of the plane is ______ 

Find the vector equation of the line passing through the point having position vector `4hat i - hat j + 2hat"k"` and parallel to the vector `-2hat i - hat j + hat k`.

Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Find acute angle between the lines `(x - 1)/1 = (y - 2)/(-1) = (z - 3)/2` and `(x - 1)/2 = (y - 1)/1 = (z - 3)/1`

Find the cartesian equation of the plane passing through the point A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.